Bayesian inference of marginal likelihood using ABC

Question

I have the following situation: suppose data $D = \{x_i\}$ iid are generated through some process with density function $f(x_i | \alpha, \beta)$ (which I think will be negative binomial) and we'd like a posterior on $\alpha$:

$f(\alpha | D) \propto \pi(\alpha) \; f(D|\alpha)=\pi(\alpha)\int d\beta \; \pi(\beta) \; \Pi_i\; f(x_i|\alpha,\beta)$

where the integral over $\beta$ is analytically intractable. What are the options here?

I've thought about MCMC sampling where I estimate the likelihood for each proposed $\alpha$ through numerical integration over $\beta$, but I'm worried that this will be prohibitively time consuming (especially since $f(x_i|\alpha,\beta)$ needs marginalised over a couple of other parameters).

The other option I've read about is approximate bayesian computation, and I'm wondering if it really is as simple as drawing an $\alpha$ from $\pi(\alpha)$ and similarly for $\beta$ then simulating $D$?

Answer 1

Answer in comments by @Juho Kokkala :
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I am not sure if I understand your question - perhaps you could just do MCMC sampling from p(α,β∣D) and discard the βs? See answers to stats.stackexchange.com/questions/125570 stats.stackexchange.com/questions/134027 ?

"Thank you! It hadn't occurred to be that by sampling from the joint density you could very cheaply find the marginal. I think this solves it. "